Excerpt from The Encyclopædia Britannica, Vol. 13: A Dictionary of Arts, Sciences, and General Literature

Mathematicians have differed as to the best way of expounding the principles of the calculus. Newton, as has already been stated, employed the theory of motion as the means of connecting its doctrines with the principles of ordinary analysis. Leibnitz, again, with the same view, conceived quantity as passing from one degree of magni tude to another by the continual addition of infinitely small parts. The mind finds no great dificnlty in dis tinctly apprehending the subject in either way. Objec tions have, however, been taken to both, and attempts made to substitute a better. Euler considered the infinitely small quantities of Leibnits es absolutely zeros, that have to each other ratios derived from those of the vanishing quantities which they replace. D'alembert proposed to make the basis of the calculus the consideration of the ratios of the limits of quantities. This method, as was indeed stated by d'alembert, does not difier in any material respect from Newton's prime and ultimate ratios. An English mathematician, Landon, substituted for the Newtonian method of fiuxions another purely algebraical. His views are contained in a work entitled The Residual Analysis, a new brands of the Algebraic Art Lagrange, too, in the Memoirs of the Berlin Academy for 177 2, proposed to base the calculus alto gether on the expansion of functions, and thus to establish it on algebraical principles merely. He subsequently developed his method in his Thor-is doe Fondionr Analyliquss and in his cons sur ls Calcul des Fondions Lagrange, however, adopted the indai tesimal method as the basis of his most important work, via, the Héeunique Analytique. He states in his preface to its second edition (1811) that when we have properly conceived the spirit of the infinitesimal method, and are convinced of the exactness of. Its results by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of abridging and eimyrlifying our demonstrations.

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